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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p><dfn class="terminology">Theorem</dfn> If <span class="process-math">\(x=x_0\)</span> is an ordinary point of</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
P(x) y^{\prime \prime}+Q(x) y^{\prime}+R(x) y=0,
\end{equation*}
</div>
<p class="continuation">then the general solution of this equation is given by</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation}
y=\sum_{n=0}^{\infty} a_n (x-x_0)^n=a_0 y_1(x)+a_1 y_2(x),\tag{5.3.1}
\end{equation}
</div>
<p class="continuation">where <span class="process-math">\(a_0\)</span> and <span class="process-math">\(a_1\)</span> are arbitrary constants and <span class="process-math">\(y_1(x)\)</span> and <span class="process-math">\(y_2(x)\)</span> are two linear independent solutions. Further, the radius of convergence for each of series solutions <span class="process-math">\(y_1\)</span> and <span class="process-math">\(y_2\)</span> is at least as large as the minimum of the radius of convergence of the series for <span class="process-math">\(Q(x)/P(x)\)</span> and <span class="process-math">\(R(x)/P(x)\text{.}\)</span></p>
<span class="incontext"><a href="sec5_3.html#p-213" class="internal">in-context</a></span>
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